# Polar Form Calculator

Created by Anna Szczepanek, PhD
Reviewed by Komal Rafay
Last updated: Jan 18, 2024

Omni's polar form calculator can help whenever you want to convert a complex number from rectangular to polar form. Scroll down to find a short article explaining what the different forms of a complex number are all about and what the polar form conversion formulas look like. Believe it or not, converting to polar form can be fun!

## What are the forms of a complex number?

The two main forms of a complex number are polar and rectangular. Usually, you first encounter the rectangular form z = a + bi, where a and b are the real part and imaginary part of z, respectively. They are both real numbers!

The polar form describes z by r × exp(φi), where:

• The magnitude r is the distance from the origin (0,0) to z; and
• The argument φ is the angle between the real axis (x-axis) and the radius connecting the origin (0,0) and z.

Look at the picture to better understand what the polar form is:

## How do I convert from rectangular to polar form?

To convert a complex number from the rectangular a + bi to polar form z, we use the formulas:

r = √(a² + b²)

and

φ = atan2(b, a).

If a > 0, then atan2(b, a) = arctan(b / a). However, if a < 0, then atan(b /a) does not give us the correct angle - it must be corrected by ±π to send us into the correct quadrant of the plane. Formally atan2(b, a) is defined as:

• atan(b / a) if a > 0;
• atan(b / a) + π if a < 0 ≤ b;
• atan(b / a) - π if a,b < 0;
• π/2 if a = 0 < b;
• -π/2 if b < 0 = a; and
• remains undefined if x = y = 0.

These are exactly the formulas behind Omni's polar form converter.

## How to use this polar form calculator?

To use this polar form converter, take a complex number in the rectangular form a + bi and input a and b into the respective fields of our tool. The two ingredients of the polar form will appear immediately in their fields: the magnitude r and phase (argument) φ. Take them and write down the polar form r × exp(iφ) of your number.

This can't get any easier!

## Omni tools for complex numbers

Complex numbers can find you in sooo many areas of science! Omni is well aware of that - we have built a whole collection of tools addressing different problems related to complex numbers. Make sure to check out at least a few of the following tools:

## FAQ

### How do I convert from trigonometric to polar form?

To convert a complex number from trigonometric to polar form, you need to:

1. Make sure your number is written as z = r × cos(φ) + r × i × sin(φ)].
2. Extract the magnitude r and the argument φ.
3. Write down your number as r × exp(φi).
4. If in doubt, use an online polar form calculator.

### What is the polar form of -1?

The answer is exp(πi). To derive this answer, observe that the modulus (magnitude) of -1 is equal to 1. Next, we must determine φ such that cos(φ) = -1 and sin(φ) = 0. We easily verify that φ = π satisfies these conditions. Hence, the whole number reads 1 × exp(πi), as claimed.

Anna Szczepanek, PhD
Rectangular form a + bi
Real part (a)
Imaginary part (b)
Polar form r × exp(iφ)
Magnitude (r)
Phase (φ)
deg
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